A set of synchronization edges, S, is sufficient if it is the minimal set such that the transitive closure of S with the program order determines all of the happens-before edges in the execution. This set is unique.
Why is the sufficient set unique?
How can we determine which synchronization edges are in the sufficient set and which aren't?
I found the answers.
According to the JLS happens-before is a strict partial order:
The happens-before order is given by the transitive closure of synchronizes-with edges and program order. It must be a valid partial order: reflexive, transitive and antisymmetric.
That means happens-before can be represented as a directed acyclic graph (DAG):
Strict partial orders correspond directly to directed acyclic graphs (DAGs)
According to the wiki:
The transitive reduction of a DAG is the graph with the fewest edges that has the same reachability relation as the DAG. It has an edge
u → vfor every pair of vertices(u, v)in the covering relation of the reachability relation ≤ of the DAG. It is a subgraph of the DAG, formed by discarding the edgesu → vfor which the DAG also contains a longer directed path fromutov. Like the transitive closure, the transitive reduction is uniquely defined for DAGs.
This "transitive reduction" is "the sufficient minimal set" in the question.
From the wiki:
It is a subgraph of the DAG, formed by discarding the edges
u → vfor which the DAG also contains a longer directed path fromutov.